Uniform convergence of a quotient

[δοτ]

Homework Statement

Let f,g be continuous on a closed bounded interval [a,b] with |g(x)| > 0 for all x in [a,b]. Suppose that $f_n \to f$ and $g_n \to g$ uniformly on [a,b]. Prove that $\frac{1}{g_n}$ is defined for large n and $\frac{f_n}{g_n} \to \frac{f}{g}$ uniformly on [a,b]. Show that this is not true if [a,b] is replaced with (a,b).

Homework Equations

The Attempt at a Solution

The fact that $g_n \to g$ uniformly coupled with |g(x)| > 0 is enough for $g_n \neq 0$ for large enough n, which means that $\frac{1}{g_n}$ is defined. I'm stuck on the other part. The fact that it is seemingly not true for an open interval domain suggests that I need that the limit functions are bounded, but I've not read anything that says a continuous limit implies continuous sequence elements, even under uniform convergence. The converse is of course true, but I'm not sure about this direction of implication. I'm really just stuck on even where to begin trying to prove that $|\frac{f_n}{g_n} - \frac{f}{g}| < \epsilon$.

Any help whatsoever is greatly appreciated.

 

[LCKurtz]

You are going to need more than $g_n(x)\ne 0$ for n large. Can you show there is $d>0$ such that $g_n(x)\ge d > 0$ for large n? You need that to show $1/g_n(x)$ is bounded. That should help.